This question is inspired by this puzzle.
You have an $m \times n$ chessboard with $m \le n$. Alice and Bob alternately move the rook (horizontally or vertically, through any number of squares). As the rook moves, it leaves a trail of painted squares: every square through which the rook passes or at which the rook stops is painted red. The rook is not allowed to pass through a red square nor can it stop on a red square. The first player who cannot move loses the game.
As seen in the original version of this puzzle, if the rook starts in a corner, the first player (Alice) can always force a win. But what if the rook does not start in the corner?
Are there any sized boards and starting positions in which the second player (Bob) can force a win?
Here is an example of the $n=m=3$ board where the rook starts in the middle. Alice can force a win by always moving 1 square until she can take all the remaining squares.
It is easy to see that the other starting squares are also winnable by Alice, thus, all squares in the $n=m=3$ board are winnable by Alice.
Note: I don't know the answer to this one. I've ruled out some initial sized boards, but a general proof for Alice being able to win all, or a counter example where Bob can win has thus far eluded me. I thought I had a simple solution, but then found a counter example for my strategy. I will have to think on it some more.
Note 2: I figured it out. Wasn't that difficult in the end after all. I will post it as an answer in a week or two if someone hasn't answered by then.